FizzBuzz in the type system

fizzbuzz.scala

// Searching for large ModAux instances can be extremely slow. 
// See this version for a faster solution: https://gist.github.com/4108026

import shapeless._, Nat._

trait FizzBuzz[N <: Nat] {
  def steps: List[String]
  def show = println(steps.reverse.mkString("\n"))
}

implicit object oneFizzBuzz extends FizzBuzz[_1] {
  def steps = toInt[_1].toString :: Nil
}

implicit def nFizzBuzz[N <: Nat](implicit
  nf: FizzBuzz[N],
  ti: ToInt[Succ[N]],
  m3: ModAux[Succ[N], _3, _0] = null,
  m5: ModAux[Succ[N], _5, _0] = null
) = new FizzBuzz[Succ[N]] {
  def steps = (
    if (m3 != null && m5 != null) "FizzBuzz" else
    if (m3 != null) "Fizz" else
    if (m5 != null) "Buzz" else toInt[Succ[N]].toString
  ) :: nf.steps
}

type _100 =
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[_22]]]]]]]]
  ]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

implicitly[FizzBuzz[_100]].show